We are not 100% sure of who wrote the four gospels. I believe we can conclude with virtually 100% certainty that Luke wrote Luke and that John wrote John. Even these cannot ever be “proven.” As for Matthew and Mark, these two gospels have been attributed to the apostle Matthew and the disciple Mark (John Mark of Acts and of Paul’s letters) since the very earliest time in the history of the church. The church fathers unanimously attributed these books to Matthew and to Mark from the earliest time. Does this prove that these two wrote the books? The answer is that it does not. However, it is far more likely that early church fathers such as Justin Martyr and Irenaeus would know who wrote these books than scholars writing today. These faithful disciples, both of whom knew disciples of Jesus who actually met the Apostle John, would be in an excellent position to tell us who wrote these books. Is this proof? I would say that we can be virtually certain that Luke wrote Luke and John wrote John, and fairly confident (but not certain) that Matthew wrote Matthew and Mark wrote Mark.
As you point out, the books themselves do not name their author, but Luke and John, for all practical purposes do identify themselves. This is one of many reasons to conclude that they wrote these books. But, as I said, the internal evidence that Matthew wrote Matthew and Mark wrote Mark is relatively weak. We must rely on the early church for this identification.
What is important to the case for Christianity is whether these gospels are inspired by God, not who wrote them. All scripture is inspired by God (2 Timothy 3:16). Of course, quoting 2 Timothy does not actually prove anything, but I believe the case for the inspiration of the Bible is so overwhelming, that, in my opinion, it is proved beyond any reasonable doubt. What is essential is that the gospels are inspired by God, now who wrote them. The four gospels have been received as inspired and have the marks of inspiration. Bottom line, these four gospels are the Word of God. The identity of the authors is a secondary (although important) question. If you heard someone saying that there is no question of the authorship of the four gospels, you should understand that that person is mistaken.
So, here is where we are: John was written by an eye-witness of the events almost for certain. Luke was written by a careful historian who was NOT an eyewitness. However, Luke is inspired by God. This we know both by faith, and by the evidence. Matthew was almost certainly (but not definitely) written by the eye-witness Matthew and Mark was almost certainly (but not definitely) written by Mark who was an eye-witness of many of the events, and a very close associate of Peter who was an eye-witness of the events recorded. Add to this the “fact” that these books are all inspired by God, and any remaining uncertainty about the author becomes insignificant to Christian belief.
Why did the writers not identify themselves? The reason is that this was standard practice in the ancient world. It may seem strange to us, but in the ancient world it was common that authors did not sign their work. This is a matter of writing style, which was common practice in the ancient Near East. It is not what we would do, but it is what writers back then very commonly did. If you read Josephus, you will see that he does the same thing. He writes about himself, but does not say “I” or “we.” This was common practice, and it explains why the biblical authors of the gospels (other than Luke) did not identify themselves.